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If $a_1 > 0$ and $$a_{n+1} = \ln\left[\frac{\exp(a_n)-1}{a_n}\right]$$ then what is the value of $$a_1 + a_1 a_2 + a_1 a_2 a_3 + a_1 a_2 a_3 a_4 + \cdots?$$

I have proved that this sequence is a decreasing one and converging to zero, but can't figure out what to do next. Any leads appreciated.

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  • $\begingroup$ Please use MathJax to type your mathematical expressions; here is a tutorial:math.meta.stackexchange.com/questions/5020/… As written it is difficult to understand your question. $\endgroup$
    – Math1000
    Dec 2, 2019 at 1:08
  • $\begingroup$ What exactly do you mean by "value of ..."? If the infinite series converges, then it has a finite real value. Is that sufficient to get what you want to prove? The value clearly is a function of $a_1$. Do you want an answer in "closed form"? $\endgroup$
    – Somos
    Dec 2, 2019 at 2:58
  • $\begingroup$ @Somos Will it blow your mind if I tell you the answer is $e^{a_1} - 1$? $\endgroup$
    – WhatsUp
    Dec 2, 2019 at 3:14
  • $\begingroup$ If that is true, then please mention that in your question. $\endgroup$
    – Somos
    Dec 2, 2019 at 3:17
  • $\begingroup$ @Somos That's not my question, rather my answer. $\endgroup$
    – WhatsUp
    Dec 2, 2019 at 3:18

1 Answer 1

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Writing $b_n = e^{a_n} - 1$, we have $b_{n + 1} + 1 = b_n / a_n$, or $b_n = a_n + a_n b_{n + 1}$.

By induction, we have $b_1 = a_1 + a_1a_2 + \dotsc + a_1a_2\dotsc a_n + a_1a_2\dotsc a_nb_{n + 1}$ for any $n \geq 1$.

Letting $n$ tends to infinity, and noting that $\lim\limits_{n\rightarrow \infty} a_n = \lim\limits_{n\rightarrow \infty}b_n = 0$, we see that the infinite sum converges to $b_1 = e^{a_1} - 1$.

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